Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 841261, 6 pages
doi:10.1155/2012/841261
Research Article
Improved Bounds for
Restricted Isometry Constants
Jiabing Ji and Jigen Peng
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
Correspondence should be addressed to Jiabing Ji, jijiabing52@163.com
Received 27 February 2012; Accepted 9 May 2012
Academic Editor: Jinde Cao
Copyright q 2012 J. Ji and J. Peng. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The purpose of this paper is to establish improved bounds for restricted isometry constants δk .
Our results, to some extent, improve and extend the well-known bound δk < 0.307 in Cai et al.,
2010 to δk < 0.308.
1. Introduction
Consider the following equation:
y Aβ z,
1.1
where the matrix A ∈ Rn×m n < m and z ∈ Rn is a vector of measurement errors. If z 0,
then 1.1 is an underdetermined linear system with fewer equations than unknowns. The
task is to reconstruct the signal β ∈ Rm based on the matrix A and the vector y. Usually, we
consider 0 minimization problem:
minβ ,
n
β∈R
0
subject to y Aβ z and z2 ≤ ,
1.2
where · 0 denotes the 0 -norm of a vector, that is, the number of its nonzero components.
2
Discrete Dynamics in Nature and Society
We need to solve this problem and find the sparsest solution among all the possible
solutions. But it requires a combinatorial search and remains an NP-hard problem that cannot
be solved in practice. Naturally, an alternative strategy is to find 1 minimization problem:
minβ ,
n
β∈R
1
subject to y Aβ z and z2 ≤ ,
1.3
and we expect to find the sparsest solution.
In order to exactly recover the sparsest solution in 1 minimization, Candès and Tao 1
introduced restricted isometry property see Restricted Isometry Constants in Definition 2.1.
So far, there are various
√ methods 1–9 to give the sufficient conditions on δ2k : Candès 3
established that δ2k < 2 − 1 ≈ 0.4142 is the sufficient condition of exactly recover k-sparse
condition was
vectors via 1 minimization a√vector x is k-sparse if x0 ≤ k. This sufficient
√
later improved to δ2k < 23 − 2/7 ≈ 0.4531 in 6 and to δ2k√< 3/4 6 ≈ 0.4652 in 5.
in 10 for the
Later, the sufficient condition was improved to δ2k < 1/1 1.25 ≈ 0.4721
√
special case that k is a multiple of 4 or k is very large and to δ2k < 4/6 6 ≈ 0.4734 in 5.
Naturally, we want to give the sufficient condition about δk . To the best of our knowledge, T.
T. Cai et al. 2 firstly show that the restricted isometry constant δk of A satisfies δk < 0.307 for
general k, then k-sparse signals are guaranteed to be recovered exactly via 1 minimization.
Based on this motivation, we construct a different partition of {1, 2, . . . , m} and then discuss
the error between original signal β and recover signal β in 1.3. The main work of this paper is
to improve the condition to δk < 0.308 and to prove that the k-sparse signals can be recovered
exactly via 1 minimization in no noise case and be estimated stably under the perturbation
of noise.
To state our main results, we firstly give the following preliminaries.
2. Preliminaries
In 2005, Candès and Tao 1 firstly present the definition of the restricted isometry constant.
Definition 2.1 see 1, restricted isometry constants. Let F be the matrix with finite collection
of vectors vj j∈J ∈ Rn as columns. For every integer 1 ≤ S ≤ |J|, the S-restricted isometry
constants δS are defined as the smallest quantity such that FT obeys
1 − δS c22 ≤ FT c22 ≤ 1 δS c22
2.1
for all subsets T ⊂ J of cardinality at most S and all real coefficients cj j∈T . Similarly, we
define the S,S’-restricted orthogonality constants θS,S for SS ≤ |J| to be the smallest quantity
such that
FT c, F c ≤ θS,S c · c 2
T
2
holds for all disjoint sets T, T ⊆ J of cardinality |T | ≤ S and |T | ≤ S .
2.2
Discrete Dynamics in Nature and Society
3
In addition, we can easily check the following monotone properties:
δ k ≤ δ k1 ,
θk,k ≤ θk1 ,k1 ,
if k ≤ k1 ≤ n,
if k ≤ k1 , k ≤ k1 , k1 k1 ≤ n.
2.3
Apart from the above relationship, Candès and Tao 1 proved that the restricted
orthogonality constant θk,k and the restricted isometry constant δk are related by the following lemma.
Lemma 2.2 see 1. One has θS,S ≤ δSS ≤ θS,S maxδS , δS for all S, S .
In the sequel, a useful inequality between 1 -norm and 2 -norm will be introduced.
Proposition 2.3 see 2. For any x ∈ Rn ,
√ x1
n
x2 − √ ≤
max|xi | − min |xi | .
1≤i≤n
4 1≤i≤n
n
2.4
At the last of preliminaries, we introduce the square root lifting inequality 10.
Lemma 2.4 see 10. For any a ≥ 1 and positive integers k, k such that ak is an integer, then
θk,ak ≤
√
aθk,k .
2.5
3. Improved Bounds for Restricted Isometry Constants
In this section, we discuss the new restricted isometry constant δk for sparse signal recovery
via 1 minimization in 1.3.
Theorem 3.1. Suppose β is k-sparse. Then the 1 minimizer β defined in 1.3 satisfies
√
2 2 1 δk
,
β − β ≤
2
1 − 13/4δk
3.1
where δk is the k-restricted isometry constant of A in 1.3.
Proof. Let h β − β ∈ Rm . Partition {1, 2, . . . , m} into the following sets:
T0 {1, 2, . . . , k},
T1 k 1, . . . , k k
,
2
T2 k
k
1, . . . , 2k , . . . ,
2
3.2
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Discrete Dynamics in Nature and Society
where k is an even number. And rearranging the indices if necessary, |h1| ≥ |h2| ≥ · · · ,
where |hi|, i 1, 2, . . . , m is the ith entry of the above vector by rearranging the indices.
Then by Proposition 2.3, we obtain
i≥1
hTi 2 ≤
1
k/2 i≥1
hTi 1 k k/2
··· .
|hk 1| − h k 4
2 3.3
By the triangle inequality for · 1 , we have
β − −hT ≤ β hT .
0 1
0 1
1
Since T0
3.4
T0c ∅, we have
β − hT c ≤ β h
c β h β
h
h
≤ β1 .
T
T
T
0
0
1
1
1
0
0
1
1
1
3.5
The last inequality holds because β solves 1.3. Then the result is that
c
hT0 ≤ hT0 1 .
3.6
1
Substituting 3.6 into 3.3, we get
i≥1
hTi 2 ≤
≤
≤
1
k/2
1
k/2
1
c
hT0 k/2
|hk 1|
4
hT0 1 k/2 hT0 2
· √
4
k
1
·
3.7
1
khT0 2 √ hT0 2
4 2
k/2
√
9 2
≤
hT0 2 .
8
And note that
| Ah, AhT0 | ≥ | AhT0 , AhT0 | −
i≥1
| AhTi , AhT0 |.
3.8
From 2.2 and 2.5 in Lemma 2.4, we have
| AhTi , AhT0 | ≤ θk/2,k hTi 2 · hT0 2 .
3.9
By Lemma 2.2, we have
θk/2,k θk/2,2·k/2 ≤
√
2δk/2k/2 √
2δk .
3.10
Discrete Dynamics in Nature and Society
5
From 3.7–3.10, we have
| Ah, AhT0 | ≥ 1 − δk hT0 22 − θk/2,k hT0 2
i≥1
hTi 2
√
√
≥ 1 − δk hT0 22 − 2δk hT0 2 · 9 2/8hT0 2
13δk
≥ 1−
hT0 22 .
4
3.11
From 1.3, we have
Ah2 A β − β ≤ Aβ − y Aβ − y2 ≤ 2.
2
2
3.12
In addition, we obtain the following relation by simple calculation
2 c
hT0 |hk 1|2 |hk 2|2 · · ·
2
≤ max|hi| · |hk 1| |hk 2| · · · i≥k1
max|hi| · hT0c i≥k1
≤
hT0 1
k
3.13
1
· hT0c .
1
Since hT0c 1 ≤ hT0 1 , we have
2 h 2
T0 1
c
.
hT0 ≤
2
k
3.14
By the norm inequality hT0 21 ≤ khT0 22 and 3.14, we have
2
c
hT0 ≤ hT0 22 .
2
3.15
From 3.7, 3.11-3.12, and 3.15, we have
√
√
2Ah2 · AhT0 2
2| Ah, AhT0 |
≤
h2 ≤ 2hT0 2 ≤
1 − 13δk /4hT0 2
1 − 13δk /4hT0 2
√
√
2 · 2 · 1 δk hT0 2 2 2 1 δk
.
≤
≤
1 − 13δk /4
1 − 13δk /4hT0 2
√
3.16
Remark 3.2. If 0, it is the case where the k-sparse signals are guaranteed to be recovered
exactly via 1 minimization under no noise situation.
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Discrete Dynamics in Nature and Society
Corollary 3.3. Let y Aβ z with z2 ≤ . Suppose β is k-sparse with k > 1. Then under the
condition δk < 0.308 the constrained 1 minimizer β given in 1.3 satisfies
β − β ≤
2
3.2344
.
0.308 − δk
3.17
Proof. The proof of this corollary can be easily obtained if we put δk < 0.308 into the inequality
3.1 in Theorem 3.1.
Acknowledgment
This work was supported by the National Science Foundation of China under contacts no.
11131006 and no. 60970149.
References
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